Noncollinear Equilibrium Points in CRTBP with Yukawa-Like Corrections to Newtonian Potential under an Oblate Primary Model

Idrisi, M. Javed; Haruna, Sunusi; Eshetie, Teklehaimanot

doi:https://doi.org/10.1155/2023/4932794

Advances in Astronomy

On this page

Abstract Introduction Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 4932794 | https://doi.org/10.1155/2023/4932794

Noncollinear Equilibrium Points in CRTBP with Yukawa-Like Corrections to Newtonian Potential under an Oblate Primary Model

M. Javed Idrisi,¹Sunusi Haruna,²and Teklehaimanot Eshetie¹

Academic Editor: Yue Wang

Received27 Feb 2023

Revised05 Jun 2023

Accepted07 Jun 2023

Published15 Jun 2023

Abstract

This study is about the effects of Yukawa-like corrections to Newtonian potential on the existence and stability of noncollinear equilibrium points in a circular restricted three-body problem when bigger primary is an oblate spheroid. It is observed that ∂x₀/∂λ = 0 = ∂y₀/∂λ at λ₀ = 1/2, so we have a critical point λ₀ = 1/2 at which the maximum and minimum values of x₀ and y₀ can be obtained, where λ ∈ (0, ∞) is the range of Yukawa force and (x₀, y₀) are the coordinates of noncollinear equilibrium points. It is found that x₀ and y₀ are increasing functions in λ in the interval 0 < λ < λ₀ and decreasing functions in λ in the interval λ₀ < λ < ∞ for all α ∈ α⁺. On the other hand, x₀ and y₀ are decreasing functions in λ in the interval 0 < λ < λ₀ and increasing functions in λ in the interval λ₀ < λ < ∞ for all α ∈ α⁻, where α ∈ (−1, 1) is the coupling constant of Yukawa force to gravitational force. The noncollinear equilibrium points are found linearly stable for the critical mass parameter β₀, and it is noticed that ∂β₀/∂λ = 0 at = 1/3; thus, we got another critical point which gives the maximum and minimum values of β₀. Also, ∂β₀/∂λ > 0 if 0 < λ < and ∂β₀/∂λ < 0 if < λ < ∞ for all α ∈ α⁻, and ∂β₀/∂λ < 0 if 0 < λ < and ∂β₀/∂λ > 0 if < λ < ∞ for all α ∈ α⁺. Thus, the local minima for β₀ in the interval 0 < λ < can also be obtained.

1. Introduction

The general three-body problem deals with the motion of three arbitrary spherically symmetric bodies considered as a point mass. The motions of these bodies are related to the Newtonian force of gravity, which are superimposed on each other and have no specific path. The closed form of analytical solution to the general three-body problem is yet to be determined.

The restricted three-body problem (R3BP) is an approximation of the general three-body problem in which one body is treated as having an infinitesimal mass compared to the other two bodies. The bigger bodies are called primaries which revolve around their common center of mass in circular or elliptical orbits in a rotating coordinate system in which the infinitesimal mass also moves without disturbing the motion of the primaries. The restricted three-body problem has five equilibrium points, three collinear, and two noncollinear or triangular. The collinear equilibria are unstable for all values of mass parameter but triangular equilibria are stable for a critical mass parameter μ₀ = 0.03852 [1].

The restricted three-body problem has been studied by many researchers in last two decades in different aspects. In the classical restricted three-body problem, the primaries are assumed as spherical in shape, but in real situation, several heavenly bodies such as Earth, Saturn, and Jupiter are sufficiently oblate. The oblateness effect in the restricted three-body problem has been investigated by El-Shaboury [2]; Khanna and Bhatnagar [3]; Raheem and Singh [4]; Ammar et al. [5]; Idrisi and Taqvi [6, 7], Singh and Umar [8]; Bury and McMahon [9]; Saeed and Zotos [10]; Alrebdi et al. [11] etc.

New theories in the contemporary world predict improvements to the theory of gravity. The Yukawa potential was first proposed by Yukawa [12] to modify the Newtonian potential. The strong interactions between particles are well described by the Yukawa potential, a nonrelativistic potential. In a two-body problem, the modified potential energy may be used to express the gravity effects on the secondary primary m in the presence of the Yukawa correction [13] aswhere V_N (r) is the Newtonian potential between the two bodies m and M, V_Y (r) is the Yukawa correction to the Newtonian potential, r is the distance between m and M, G is the Newtonian gravitational constant, α ϵ (−1, 1) is the coupling constant of the Yukawa force to the Gravitational force, and λ ϵ (0, ∞) is the range of the Yukawa force [14]. Therefore, the corresponding force between m and M can be expressed as

As α ⟶ 0, the Newtonian gravitational force can be obtained.

In the restricted three-body problem, Kokubun [14] has included Yukawa-like corrections to Newtonian potential. His findings differed significantly from the purely Newtonian case. Reference [15] provides the minimal values of the Yukawa coupling constant for the artificial satellites LAGEOS and LAGEOS II. Massa [16] investigated Mach’s principle and Yukawa potential within the Sciama linear approach framework. Haranas and Ragos [17] investigated satellite dynamics while taking Yukawa-like corrections into account. Pricopi [18] has investigated the stability of celestial orbits under the effect of the Yukawa potential in the two-body problem. Reference [19] has analyzed the elliptical and circular orbits of the Earth while taking into account the Yukawa potential and Poynting–Robertson effect. The dynamics and stability of the two-body problem were examined by Cavan et al. [20] while taking the Yukawa corrections to Newtonian potential into account. Idrisi et al. [21] have investigated the triangular equilibria in the framework of Yukawa correction to Newtonian potential in the circular restricted three-body problem.

The dynamics surrounding noncollinear equilibrium points in a circular restricted three-body problem with a Yukawa-like adjustment to Newtonian potential under an oblate primary model piqued our attention. The existence and linear stability of noncollinear equilibrium points under an oblate primary model with Yukawa like-corrections to Newtonian potential have been examined in this study.

2. Yukawa Correction to Newtonian Potential

The modified potential between two bodies M and m can be described as follows:where V_N (r) = Newtonian potential between the two bodies M and m, V_Y (r) = Yukawa correction to the Newtonian potential, r = distance between m and M, G = Newtonian gravitational constant, α ∈ (−1, 1) is the coupling constant of Yukawa force to the gravitational force, and λ ∈ (0, ∞) is the range of Yukawa force [14].

Therefore, the corresponding force between M and m can be expressed aswhere F_N (r) = Newtonian gravitational force between M and m and F_Y (r) = Yukawa correction to Newtonian gravitational force between M and m.

From (4), as α ⟶ 0 or λ ⟶ 0, the term F_Y (r) vanishes and F(r) = F_N(r). If α < 0, F(r) < F_N(r) and for α > 0, F(r) > F_N(r), Figure 1. Thus, as α increases in the interval (−1, 1), the force between m and M also increases and vice-versa.

But as λ ⟶ ∞ the force between M and m is given by

From (5), it is clear that as α ⟶ −1, F_∞(r) ⟶ 0, i.e., the force between m and M reduces as α reduces. For α ⟶ 0, F_∞ (r) ⟶ F_N (r) and the Newtonian gravitational force can be obtained. But as α ⟶ 1, F_∞ (r) ⟶ 2F_N (r), i.e., the force acting between m and M is twice of the Newtonian gravitational force, as shown in Figure 2.

3. Model Description and Equations of Motion

Let us consider two primaries P₁ and P₂ having masses m₁ and m₂ (m₁ > m₂) moving around their common center of mass in circular orbits. The more massive primary m₁ is considered to be an oblate body while less massive primary m₂ is spherical in shape. The equations of motion of the infinitesimal mass in a barycentric synodic co-ordinate system (x, y) and dimensionless variables areand the potential function U can be expressed asσ = ()/5r² is the oblateness factor due to bigger primary m₁, r_e and r_p are the equatorial and polar radii respectively of m₁, r is the distance between m₁ and m₂ considered as unity, n is the mean-motion of the primaries, and defined as

|α| < 1 is the coupling constant of Yukawa force to gravitational force, λ ∈ (0, ∞) is the range of Yukawa force.

We can define a mass parameter β > 0 as

Therefore, the distances of infinitesimal mass from the primaries P₁ and P₂, are given by

The Jacobi integral associated to the problem is given bywhere is the velocity of infinitesimal mass and C is Jacobi constant.

4. Noncollinear Equilibrium Points

The noncollinear equilibrium points are the solution of the equations U_x = 0 and U_y = 0, y ≠ 0, i.e.,

On eliminating r₁ and r₂ from the (12) and (13), respectively, we have

The solution of (15) is r₁ = 1. To solve (14), we assume that r₂ = 1 + δ, δ << 1. On substituting r₂ = 1 + δ in (14) and considering only linear terms in α and δ and then solving it for δ, we obtain

Now, solving r₁ = 1 and (17), we have the coordinates of noncollinear equilibrium points E_{4, 5} (x₀, y₀), i.e.,

For a nonoblate case, i.e., σ = 0 we obtain r_i = 1 which is the classic case of restricted three-body problem [1], and hence in the nonoblate case, the noncollinear equilibrium points are not affected by the Yukawa force [21]. For α = 0, the results are agreed with [22].

As shown in Figure 3, f(α, λ) is a continuous function for |α| < 1 and λ ∈ (0, ∞), and . Thus, for very small and large values of λ, the term f(α, λ) becomes unity and the noncollinear equilibrium points E_{4, 5}(, ) in this case are given by

Since ∂x₀/∂λ = 0 = ∂y₀/∂λ at λ = 1/2. So we have a critical point λ = λ₀ = 1/2 at which the maximum and minimum values of x₀ and y₀ can be obtained. As it has been examined that ∂x₀/∂λ, ∂y₀/∂λ > 0 for 0 < λ < λ₀ and ∂x₀/∂λ, ∂y₀/∂λ < 0 for λ₀ < λ < ∞ for all α ∈ α⁺. Similarly, ∂x₀/∂λ, ∂y₀/∂λ < 0 for 0 < λ < λ₀ and ∂x₀/∂λ, ∂y₀/∂λ > 0 for λ₀ < λ < ∞ for all α ∈ α⁻. Thus, it is concluded that x₀ and y₀ are increasing functions in λ in the interval 0 < λ < λ₀ and decreasing functions in λ in the interval λ₀ < λ < ∞ for all α ∈ α⁺. On the other hand, x₀ and y₀ are decreasing functions in λ in the interval 0 < λ < λ₀ and increasing functions in λ in the interval λ₀ < λ < ∞ for all α ∈ α⁻.

For α ∈ α⁺, as λ increases in the interval 0 < λ < λ₀, the abscissa x₀ of E₄ moves toward the center of mass of the system and the ordinate y₀ moves vertically upward and vice-versa. In the interval λ₀ < λ < ∞, x₀ and y₀ decrease and hence approach to and , respectively, as λ increases and vice versa. For α ∈ α⁻, as λ increases in the interval 0 < λ < λ₀, the abscissa x₀ moves away from and y₀ moves vertically downward and vice-versa. In the interval λ₀ < λ < ∞, x₀ and y₀ increase and hence approach to and , respectively, as λ increases and vice-versa (Figures 4 and 5).

The noncollinear equilibrium points E_{4, 5} at the critical point λ = 1/2 have maximum or minimum values according to α ∈ α⁺ or α ∈ α⁻, respectively, are given as

5. Stability of Equilibrium Points

The variational equations of motion can be obtained by perturbing the equilibrium point (x₀, y₀) to a small displacement (δ₁, δ₂), δ_i << 1, i = 1, 2. Therefore, on substituting x = x_o + δ₁ and y = y_o + δ₂ in Equation (3), we have

As δ_i << 1 and |α| < 1, therefore we consider only linear terms in δ₁, δ₂, and α, and the characteristic equation corresponding to (21) is given bywhere

The quadratic equation corresponding to (22) is given bywhere

The roots of (24) are

The motion near the equilibrium point (x₀, y₀) is said to be bounded if ≥ 0, i.e.,

On solving the inequality (19), we getwhere μ₀ = 0.0385209…. For α = 0, β₀ = = μ₀ − σ/√69 is the critical mass parameter in the circular restricted three-body problem when bigger primary is an oblate body [22]. For α = 0 and σ = 0, β₀ = μ₀ = 0.0385209… is the critical mass parameter in the classical circular restricted three-body problem [1]. For σ = 0, all results are in conformity with those of Idrisi et al. [21]. Thus, the noncollinear equilibrium points obtained in the proposed model are linearly stable for the critical mass parameter β₀ defined in (26).

The third term in (26) vanishes either for α = 0 or Q = 0, i.e., . Thus, β₀ > in the interval 0 < λ < λ₁ and β₀ < in the interval λ₁ < λ < ∞ for all α ∈ (−1, 0). Similarly, β₀ < when 0 < λ < λ₁ and β₀ > in the interval λ₁ < λ < ∞ for all α ∈ (0, 1), Figure 6.

From (26), we have the following observations: at λ = 1/3. Thus, λ = = 1/3 is a critical point which gives the maximum and minimum values of β₀. Also, if 0 < λ < and if < λ < ∞ for all α ∈ (−1, 0), and if 0 < λ < and if < λ < ∞ for all α ∈ (0, 1). Thus, we have the local minima in the interval 0 < λ < . The local maximum and minimum values of β₀ at the critical point λ = are given by

In Figure 7, the stability surface is plotted, and it can be seen that when α rises, the stability surface does too and vice versa. As a result, the noncollinear equilibrium points are on the surface are stable and unstable otherwise.

The shaded region in Figure 8 corresponds to stable region for the noncollinear equilibrium points, and it is seen that as alpha increases the stability region also increases and vice-versa.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

6. Real Application to the Earth-Moon System

From astrophysical data [23], mass of Earth = 5.972 ×10²⁴ kg, mass of moon = 7.348 × 10²² kg, axes of the Earth: r_e = 6378.140 km, r_p = 6356.755 km, and average distance between Earth and moon = 382500 km.

In the Earth-moon system, λ = 400000 km[24].

In a dimensionless system, we have

Table 1 lists the numerical locations of noncollinear equilibrium points E_{4, 5}(x₀, y₀) for the aforementioned values of β, σ and for |α| < 1. For all possible values of α, it has been found that the numerical values of x₀ and y₀ are identical up to six decimal places.

7. Conclusion

We studied the dynamics around noncollinear equilibrium points in the circular restricted three-body problem under the considerations of oblateness of more massive primary and Yukawa-like corrections to Newtonian potential. The modified gravitational force between the two masses M and m, therefore, can be written as F(r) = F_N(r) + F_Y(r), where F_N(r) is Newtonian gravitational force between M and m, and F_Y(r) is Yukawa correction to Newtonian gravitational force between M and m. It is found that as α ⟶ 0 or λ ⟶ 0, the term F_Y (r) vanishes and F(r) = F_N(r), where α ∈ (−1, 1) is the coupling constant of Yukawa force to gravitational force and λ ∈ (0, ∞) is the range of Yukawa force. If α < 0, F(r) < F_N(r) and for α > 0, F(r) > F_N(r), Figure 1. Thus, as α increases in the interval (−1, 1), the force between m and M also increases and vice-versa. But as λ ⟶ ∞, the force between M and m is given by F_∞(r) and F_∞(r) ⟶ 0 as α ⟶ −1, i.e., the force between m and M reduces as α reduces. For α ⟶ 0, F_∞(r) ⟶ F_N(r) and the Newtonian gravitational force can be obtained. But as α ⟶ 1, F_∞(r) ⟶ 2F_N(r), i.e., the force acting between m and M is twice of the Newtonian gravitational force, as shown in Figure 2.

The nonequilibrium points are the solutions of r₁ = 1 and (17). On solving these equations, we got two noncollinear equilibrium points E_{4, 5}(x₀, y₀) given in (18). For nonoblate case, i.e., σ = 0 we obtain r_i = 1 which is the classic case of restricted three-body problem [1], and hence in the nonoblate case, the noncollinear equilibrium points are not affected by the Yukawa force [21]. For α = 0, the results are agreed with [22]. It is observed that ∂x₀/∂λ = 0 = ∂y₀/∂λ at λ = 1/2. So, we have a critical point λ = λ₀ = 1/2 at which the maximum and minimum values of x₀ and y₀ can be obtained. As it has been examined that ∂x₀/∂λ, ∂y₀/∂λ > 0 for 0 < λ < λ₀ and ∂x₀/∂λ, ∂y₀/∂λ < 0 for λ₀ < λ < ∞ for all α ∈ α⁺. Similarly, ∂x₀/∂λ, ∂y₀/∂λ < 0 for 0 < λ < λ₀ and ∂x₀/∂λ, ∂y₀/∂λ > 0 for λ₀ < λ < ∞ for all α ∈ α⁻. Thus, it is concluded that x₀ and y₀ are increasing functions in λ in the interval 0 < λ < λ₀ and decreasing functions in λ in the interval λ₀ < λ < ∞ for all α ∈ α⁺. On the other hand, x₀ and y₀ are decreasing functions in λ in the interval 0 < λ < λ₀ and increasing functions in λ in the interval λ₀ < λ < ∞ for all α ∈ α⁻. For α ∈ α⁺, as λ increases in the interval 0 < λ < λ₀, the abscissa x₀ of E₄ moves toward the center of mass of the system and the ordinate y₀ moves vertically upward and vice-versa. In the interval λ₀ < λ < ∞, x₀ and y₀ decrease and hence approach to and , respectively, as λ increases and vice versa. For α ∈ α⁻, as λ increases in the interval 0 < λ < λ₀, the abscissa x₀ moves away from and y₀ moves vertically downward and vice-versa. In the interval λ₀ < λ < ∞, x₀ and y₀ increase and hence approach to and , respectively, as λ increases and vice-versa (Figures 4 and 5).

The noncollinear equilibrium points obtained in the proposed model are linearly stable for the critical mass parameter β₀ defined in (26). From (26), ∂β₀/∂λ = 0 at = 1/3 thus we got another critical point which gives the maximum and minimum values of β₀. Also, ∂β₀/∂λ > 0 if 0 < λ < and ∂β₀/∂λ < 0 if < λ < ∞ for all α ∈ α⁻, and ∂β₀/∂λ < 0 if 0 < λ < and ∂β₀/∂λ > 0 if < λ < ∞ for all α ∈ α⁺. The local maximum and minimum values of β₀ at the critical point λ = are given in (27).

Data Availability

The data used to support the findings of this study are included within this research article. For simulation, we have used data from other research papers which are properly cited.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

V. Szebehely, Theory of Orbits, the Restricted Problem of Three Bodies, Academic Press, New York and London, 1967.
S. M. El-Shaboury, “Equilibrium solutions of the restricted problem of 2+2 axisymmetric rigid bodies,” Celestial Mechanics and Dynamical Astronomy, vol. 50, no. 3, pp. 199–208, 1991.
View at: Publisher Site | Google Scholar
M. Khanna and K. B. Bhatnagar, “Existence and stability of libration points in the restricted three body problem when the smaller primary is a triaxial rigid body and the bigger one an oblate spheroid,” Indian Journal of Pure and Applied Mathematics, vol. 30, pp. 721–733, 1999.
View at: Google Scholar
A. Raheem and J. Singh, “Combined effects of perturbations, radiation and oblateness on the stability of equilibrium points in the restricted three-body problem,” The Astronomical Journal, vol. 131, no. 3, pp. 1880–1885, 2006.
View at: Publisher Site | Google Scholar
M. K. Ammar, S. M. El-Shaboury, and M. R. Amin, “Third-order secular solution of the variational equations of motion of a satellite in orbit around a non-spherical planet,” Astrophysics and Space Science, vol. 340, no. 1, pp. 43–61, 2012.
View at: Publisher Site | Google Scholar
M. J. Idrisi and Z. A. Taqvi, “Restricted three-body problem when one of the primaries is an ellipsoid,” Astrophysics and Space Science, vol. 348, no. 1, pp. 41–56, 2013.
View at: Publisher Site | Google Scholar
M. J. Idrisi and Z. A. Taqvi, “Existence and stability of the non-collinear libration points in the restricted three body problem when both the primaries are ellipsoid,” Astrophysics and Space Science, vol. 350, no. 1, pp. 133–141, 2014.
View at: Publisher Site | Google Scholar
J. Singh and A. Umar, “Effect of oblateness of an artificial satellite on the orbits around the triangular points of the Earth-Moon system in the axisymmetric ER3BP,” Differential Equations and Dynamical Systems, vol. 25, no. 1, pp. 11–27, 2017.
View at: Publisher Site | Google Scholar
L. Bury and J. McMahon, “The effect of zonal harmonics on dynamical structures in the circular restricted three-body problem near the secondary body,” Celestial Mechanics and Dynamical Astronomy, vol. 132, no. 9, p. 45, 2020.
View at: Publisher Site | Google Scholar
T. Saeed and E. E. Zotos, “On the equilibria of the restricted three-body problem with a triaxial rigid body – I, oblate primary,” Results in Physics, vol. 23, Article ID 103990, 2021.
View at: Publisher Site | Google Scholar
H. I. Alrebdi, F. L. Dubeibe, and E. E. Zotos, “On the equilibria of the restricted three-body problem with a triaxial rigid body, II: prolate primary,” Results in Physics, vol. 38, Article ID 105623, 2022.
View at: Publisher Site | Google Scholar
H. Yukawa, “On the interaction of elementary particles,” Proceedings of the Physico-Mathematical Society of Japan 3^rd Series, vol. 17, pp. 48–57, 1935.
View at: Google Scholar
E. Fischbach, C. Talmadge, and D. E. Krause, “Exponential models of non-Newtonian gravity,” Physical Review D, vol. 43, no. 2, pp. 460–467, 1991.
View at: Publisher Site | Google Scholar
F. Kokubun, “Restricted problem of three bodies with Newtonian+Yukawa potential,” International Journal of Modern Physics D, vol. 13, no. 05, pp. 783–806, 2004.
View at: Publisher Site | Google Scholar
N. I. Kolosnitsyn and V. N. Melnikov, “Test of the inverse square law through precession of orbits,” General Relativity and Gravitation, vol. 36, no. 7, pp. 1619–1624, 2004.
View at: Publisher Site | Google Scholar
C. Massa, “Mach’s principle and Yukawa-like corrections to the Newtonian potential as a bound for the mean energy density of the universe,” Astrophysics and Space Science, vol. 315, no. 1-4, pp. 285-286, 2008.
View at: Publisher Site | Google Scholar
I. Haranas and O. Ragos, “Yukawa-type effects in satellite dynamics,” Astrophysics and Space Science, vol. 331, no. 1, pp. 115–119, 2011.
View at: Publisher Site | Google Scholar
D. Pricopi, “Stability of the celestial body orbits under the influence of Yukawa potential,” Astrophysics and Space Science, vol. 361, no. 8, p. 277, 2016.
View at: Publisher Site | Google Scholar
I. Haranas, O. Ragos, I. Gkigkitzis, I. Kotsireas, C. Martz, and S. Van Middekoop, “The Poynting-Robertson effect in the Newtonian potential with a Yukawa correction,” Astrophysics and Space Science, vol. 363, no. 1, p. 3, 2018.
View at: Publisher Site | Google Scholar
E. Cavan, I. Haranas, I. Gkigkitzis, and K. Cobbett, “Dynamics and stability of the two body problem with Yukawa correction,” Astrophysics and Space Science, vol. 365, no. 2, p. 36, 2020.
View at: Publisher Site | Google Scholar
M. J. Idrisi, T. Eshetie, T. Tilahun, and M. Kerebh, “Triangular equilibria in R3BP under the consideration of Yukawa correction to Newtonian potential,” Journal of Applied Mathematics, vol. 2022, Article ID 4072418, 6 pages, 2022.
View at: Publisher Site | Google Scholar
R. K. Sharma and P. V. Subba Rao, “Stationary solutions and their characteristic exponents in the restricted three-body problem when the more massive primary is an oblate spheroid,” Celestial Mechanics, vol. 13, no. 2, pp. 137–149, 1976.
View at: Publisher Site | Google Scholar
K. R. Lang, Astrophysical Data: Planets and Stars, Springer-Verlag, Berlin, Germany, 1992.
J. Muller, “Lunar Laser Ranging – A Science Tool for Geodesy and General Relativity,” in Proceedings of the 16th International Conference on Laser Ranging, Poznan, Poland, 2008, https://cddis.nasa.gov/lw16/docs/presentations/sci_8_Mueller.pdf.
View at: Google Scholar

Copyright

Copyright © 2023 M. Javed Idrisi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

440

Downloads

407

Citations